Simultaneous first order reaction

Lightfoot, E. N. A.l.Ch.E.J. 4 (1958) 499. Steady state absorption of a sparingly soluble gas in an agitated tank with simultaneous first order reaction. 

Olson, D.L. and Shuman, M.S. (1983) Kinetic spectrum method for analysis of simultaneous, first-order reactions and application to copper(II) dissociation from aquatic macromolecules. Anal. Chem., 55, 1103-1107. 

Some authors describe the kinetic of the adsorption process as a linear combination of exponential terms analogous to a series of simultaneous first-order reactions. Neuman and Neuman (1958) described the kinetics of the adsorption process by the following equation ... 

Here, also the decoloration curves can be resolved by two simultaneous first - order reactions, as above for spirobenzopyrans. 

Diffusion with a convection and simultaneous first order reaction in a rectangular plate can be simulated using the program described above by using minor modifications. Consider the composition profile in a packed tube reactor undergoing isothermal linear kinetics with axial diffusion. The governing equation is... 

Figure 2. Resolution of the experimentally obtained time-conversion curve by two simultaneous first-order reactions (3).

A compound X undergoes two simultaneous first-order reactions... 

The three preceding equations may be solved simultaneously by the shooting method. A result for a first-order reaction is shown in Fig. 23-20, together with the case of uniform poisoning. 

Systems of reversible first-order reactions lead to sets of simultaneous linear differential equations with constant coefficients. A solution may be obtained by means of a matrix formulation that is widely used in quantum mechanics and vibrational... 

From this expression, it is obvious that the rate is proportional to the concentration of A, and k is the proportionality constant, or rate constant, k has the units of (time) usually sec is a function of [A] to the first power, or, in the terminology of kinetics, v is first-order with respect to A. For an elementary reaction, the order for any reactant is given by its exponent in the rate equation. The number of molecules that must simultaneously interact is defined as the molecularity of the reaction. Thus, the simple elementary reaction of A P is a first-order reaction. Figure 14.4 portrays the course of a first-order reaction as a function of time. The rate of decay of a radioactive isotope, like or is a first-order reaction, as is an intramolecular rearrangement, such as A P. Both are unimolecular reactions (the molecularity equals 1). 

Danckwerts et al. (D6, R4, R5) recently used the absorption of COz in carbonate-bicarbonate buffer solutions containing arsenate as a catalyst in the study of absorption in packed column. The C02 undergoes a pseudo first-order reaction and the reaction rate constant is well defined. Consequently this reaction could prove to be a useful method for determining mass-transfer rates and evaluating the reliability of analytical approaches proposed for the prediction of mass transfer with simultaneous chemical reaction in gas-liquid dispersions. 

Again for the case of the sparingly soluble gas whose absorption is accompanied by a simultaneous irreversible first-order reaction, Lightfoot (L5, L6) made the following assumptions ... 

A slightly different approach was taken by Gill (G15), who considered the case of a bubble moving through a stationary liquid with mass transfer accompanied by simultaneous first-order chemical reaction. His assumptions were as follows ... 

Equations (2.22) and (2.23) become indeterminate if ks = k. Special forms are needed for the analytical solution of a set of consecutive, first-order reactions whenever a rate constant is repeated. The derivation of the solution can be repeated for the special case or L Hospital s rule can be applied to the general solution. As a practical matter, identical rate constants are rare, except for multifunctional molecules where reactions at physically different but chemically similar sites can have the same rate constant. Polymerizations are an important example. Numerical solutions to the governing set of simultaneous ODEs have no difficulty with repeated rate constants, but such solutions can become computationally challenging when the rate constants differ greatly in magnitude. Table 2.1 provides a dramatic example of reactions that lead to stiff equations. A method for finding analytical approximations to stiff equations is described in the next section. 

The extension to multiple reactions is done by writing Equation (3.1) (or the more complicated versions of Equation (3.1) that will soon be developed) for each of the N components. The component reaction rates are found from Equation (2.7) in exactly the same ways as in a batch reactor. The result is an initial value problem consisting of N simultaneous, first-order ODEs that can be solved using your favorite ODE solver. The same kind of problem was solved in Chapter 2, but the independent variable is now z rather than t. 

Only numerical solutions are possible when Equation (9.24) is solved simultaneously with Equation (9.14). This is true even for first-order reactions because of the intractable nonlinearity of the Arrhenius temperature dependence. 

The Effectiveness Factor Analysis in Terms of Effective Diffusivities First-Order Reactions on Spherical Pellets. Useful expressions for catalyst effectiveness factors may also be developed in terms of the concept of effective diffusivities. This approach permits one to write an expression for the mass transfer within the pellet in terms of a form of Fick s first law based on the superficial cross-sectional area of a porous medium. We thereby circumvent the necessity of developing a detailed mathematical model of the pore geometry and size distribution. This subsection is devoted to an analysis of simultaneous mass transfer and chemical reaction in porous catalyst pellets in terms of the effective diffusivity. In order to use the analysis with confidence, the effective diffusivity should be determined experimentally, since it is difficult to obtain accurate estimates of this parameter on an a priori basis. 

No results have yet been reported for the kinetics of the pyrolysis of vinylcyclobutane though there is some indirect evidence that one of the reaction paths would yield cyclohexene. Kinetic results are available for isopropenylcyclobutane and by analogy with cyclopropane systems the behaviour of this compoimd should be very similar to vinylcyclobutane. It has been reported (Ellis and Frey, 1963) that the pyrolysis of isopropenylcyclobutane gives rise to ethylene, isoprene and 1-methyl-cyclohexene. These products arise by two simultaneous first-order processes which are both homogeneous ... 

In practice, of course, it is rare that the catalytic reactor employed for a particular process operates isothermally. More often than not, heat is generated by exothermic reactions (or absorbed by endothermic reactions) within the reactor. Consequently, it is necessary to consider what effect non-isothermal conditions have on catalytic selectivity. The influence which the simultaneous transfer of heat and mass has on the selectivity of catalytic reactions can be assessed from a mathematical model in which diffusion and chemical reactions of each component within the porous catalyst are represented by differential equations and in which heat released or absorbed by reaction is described by a heat balance equation. The boundary conditions ascribed to the problem depend on whether interparticle heat and mass transfer are considered important. To illustrate how the model is constructed, the case of two concurrent first-order reactions is considered. As pointed out in the last section, if conditions were isothermal, selectivity would not be affected by any change in diffusivity within the catalyst pellet. However, non-isothermal conditions do affect selectivity even when both competing reactions are of the same kinetic order. The conservation equations for each component are described by... 

These authors used 6 1 and 30 1 alcohol-to-oil molar ratios for both methanol and butanol. As expected, a pseudo first-order reaction was found at large excess of alcohol for both alcohols. At low excess alcohol, however, the butanolysis reaction (30°C) was second-order, but the methanolysis reaction (40°C) was reported to be a combination of a second-order consecutive reaction and a fourth-order shunt reaction. The shunt reaction, in which three methanol molecules simultaneously attack a TG molecule, was adopted to better fit the kinetic data. However, such a reaction is highly unlikely. Nureddini et al. later found that the inclusion of a shunt mechanism was not necessary to fit the kinetic data of the transesterification reaction, and Boocock et al showed that the shunt reaction assumption came as a misinterpretation of the observed kinetics. At low temperatures (20 0°C) the multiphase methanolysis reaction... 

It is useful to briefly consider a simple conceptional model that considers simultaneous inputs and outputs of a compound in an organism, the one-box approach (see Section 12.4 for a general discussion of one-box models). In this approach we assume that the organism (i.e., the fish) is a well-mixed reactor (which, of course, it is not), and we define all processes as first-order reactions. The temporal change in concentration of a given compound i in the fish, Clfish, can then be described simply by ... 

Riesenfeld and Bohnholtzer and Riesenfeld and Schumacher used ozone concentrated by liquefaction and distillation. From their kinetic measurements they conclude that a reaction of the second order and one of the first order take place simultaneously at quite low pressures, 6-60 mm. Hg the first order reaction predominates. The velocity constants of the second order reaction are not influenced by the total pressure, while those of the first order reaction appear to be inversely proportional to the total pressure. The figures given show that the first order reaction at the lower pressures is considerably influenced by the surface, and is quite probably a heterogeneous reaction, though the authors themselves do not consider this to be definitely shown. The decomposition appears to be rather sensitive to catalysts such as dust particles. 

The absorption of ozone from the gas occurred simultaneously with the reaction of the PAH inside the oil droplets. In order to prove that the mass transfer rates of ozone were not limiting in this case, the mass transfer gas/water was optimized and the influence of the mass transfer water/oil was studied by ozonating various oil/water-emulsions with defined oil droplet size distributions. No influence of the mean droplet diameter (1.2 15 pm) on the reaction rate of PAH was observed, consequently the chemical reaction was not controlled by mass transfer at the water/oil interface or diffusion inside the oil droplets. Therefore, a microkinetic description was possible by a first order reaction with regard to the PAH concentration (Kornmuller et al., 1997 a). The effects of pH variation and addition of scavengers indicated a selective direct reaction mechanism of PAH inside the oil droplets... 

Finding the time required for a particular conversion involves the solution of two simultaneous equations, i.e. 1.24 or 1.25 for the material balance and 1.27 for the heat balance. Generally, a solution in analytical form is unobtainable and numerical methods or analogue simulation must be used. Taking, for example, a first-order reaction with constant volume ... 

The influence which the simultaneous transfer of heat and mass in porous catalysts has on the selectivity of first-order concurrent catalytic reactions has recently been investigated by 0stergaard(27). As shown previously, selectivity is not affected by any limitations due to mass transfer when the process corresponds to two concurrent first-order reactions ... 

Absorbance studies show considerable overlap of the different spectral bands in metal-ethylenediamine solutions. It appears likely that the simultaneous first-order processes in reducing water and other solutes result from different reaction rates of the various species. With this assumption, and using Beer s Law for the individual species, one obtains... 

The apparent HDM reaction orders greater than unity have been attributed to the presence of more than one class of metal compounds reacting with different rates (Oleck and Sherry, 1977 Cecil et al., 1968). Just as in hydrodesulfurization, the simultaneous occurrence of several first-order reactions with different rates can lead to an apparent reaction order greater than unity (de Bruijn, 1976). Wei and Hung (1980) theoretically demonstrated conditions whereby two first-order reactions give rise to apparent second-order kinetics. 

The reactions can be described by use of two simultaneous first-order expressions one expression for easy-to-remove sulfur and a separate expression for difficult-to-remove sulfur. 

Thus, in terms of a sufficiently simple two-route mechanism, it is possible to interpret the effects observed by different authors [48, 53, 62, 98] (1) a jumpwise increase in the reaction rate at definite temperatures (2) temperature independence of the rate and simultaneously first order with respect to CO at "low T and Pco (3) zero order with respect to CO at "high T. The model corresponding to the two-route mechanism and using the parameters from ref. 49 predicts the existence of critical effects first discovered by Golchet and White [62] under deep vacuum. 

Within the pressure range 2 to 30 Torr isopropanol, the dehydrogenation is a first-order reaction with strong inhibition by water. The simultaneously occurring dehydration is a zero-order reaction (Eqs. 27 to 29). Illumination was found to produce no change in the kinetics. 

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